On the Theory of Matrix-valued Functions Belonging to the Smirnov Class
نویسنده
چکیده
A theory of matrix-valued functions from the matricial Smirnov class Nn (D) is systematically developed. In particular, the maximum principle of V.I.Smirnov, inner-outer factorization, the Smirnov-Beurling characterization of outer functions and an analogue of Frostman’s theorem are presented for matrix-valued functions from the Smirnov class Nn (D). We also consider a family Fλ = F − λI of functions belonging to the matricial Smirnov class which is indexed by a complex parameter λ. We show that with the exception of a ”very small” set of such λ the corresponding inner factor in the inner-outer factorization of the function Fλ is a Blaschke-Potapov product. The main goal of this paper is to provide users of analytic matrix-function theory with a standard source for references related to the matricial Smirnov class.
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